This is quite a soft question. I'm looking for any properties that a graph $G$ on $n$ vertices satisfying the following conditions might have:

$\chi(G)=n-2$

$|E(G)|>(n^2-3n+6)/2$

Clearly, for example, $n$ must be greater than 3, the clique number of $G$ must be less than or equal to $n-2$, and the density of the graph must be greater than: $\displaystyle\frac{n^2-3n+6}{n^2-n}$.